﻿The thesis considers Fisher market with Constant Elasticity of Substitution
(CES) utilities, including Linear utility, Leontief utility and Cobb-Douglas
utility, and studies strategic behaviors of individual buyers in market equilibria. The notion of incentive ratio is introduced to capture the extent to
which utilities can be increased by strategic behaviors of individuals. Formally, in a given market, the incentive ratio of any given buyer is defined to
be his maximum possible utility obtainable by employing strategic behaviors
divided by his utility when bidding truthfully, given any fixed bids of all
other buyers. The incentive ratio of the whole market is defined to be the
maximum incentive ratio of all buyers. Note that the ratio is at least one
which happens for an incentive compatible mechanism. While buyers may
get larger utilities when they behave strategically, it is interesting to know
the range of its improvement. This thesis shows that the benefits of strategic
behaviors may be quite limited. For Leontief market, the incentive ratio is
at most 2, and we give an example to show that this upper bound is tight.
Moreover, the incentive ratio of Leontief market is insensitive to market size.
For Linear market, the thesis presents a constructive proof for the incentive
ratio of 2-buyer market. For Cobb-Douglas market, the thesis derive the best response strategy of the buyers and the su±cient and necessary condition for bidding truthful to be a dominant strategy for all buyers. We also prove the incentive ratio is independent of the number of buyers. Meanwhile, simulation results show that the incentive ratio of Cobb-Douglas market is indeed small.
In addition, the thesis obtains a matching exponential algorithmic bound
for TUCKER in the oracle model, in all dimensions. The lower bound is
based on a reduction from the Direction Preserving Zero Point (DPZP) to
TUCKER. DPZP is a new mathematical structure introduced recently as
an appropriate technical tool to formulate the more traditional fixed point
computational problems. The upper bound requires a combinatorial parity
lemma that relates the existence of the complementary edges on the boundary
with certain parity properties of the fully-colored simplices on the boundary
of the hypergrid. The connection also allows us to derive an immediate proof
that TUCKER is PPAD-complete for all constant dimensions.